Abstract— This paper discusses the thermal behavior of a highpressure mercury lamp in a horizontal position compared to that
of a vertical lamp when the supply current varies. The model
adopted is 3-D, steady, and dc powered. After the validation of
the model, pressure of the lamp is kept at 6 × 105 Pa and the
supply current varied from 1 to 10 A. Then, by comparing the
case of the lamp in a horizontal position with that in a vertical
one, the temperature fields, the flow of heat conduction, the flow
of convective heat, and the accumulation of mercury behind the
electrodes are analyzed.
Index Terms— Convection, electric current, energy transfer,
heat conduction, high pressure, mercury discharge lamps, vertical
and horizontal positions.

I. I NTRODUCTION

T

HIS paper describes a 3-D modeling of a high-pressure
discharge plasma. The proposed model is applied to this
paper of the transfer of energy in a dc-powered mercury
discharge lamp (MDL) at constant pressure but which can be
operated in both vertical and horizontal positions.
The choice of this discharge as a medium for the application
of the model is, first, due to the valid possibilities that it offers.
In fact, it serves as a support for numerous experimental and
theoretical works [1], [2]. Then, even if the lamp is replaced
by other systems, it can continue to be extensively used.
In fact, virtually all discharge lamps contain significant
amounts of mercury (they are often the major element in the
discharge), which sometimes plays the role of the buffer gas
and at other times as the active gas or other additives.
In addition, these lamps are characterized by a high luminous efficiency and a very good color rendering index, which
is why they are used.
Most works have been devoted to the study of lamps placed
in vertical positions [3]–[9]. For example, Charrada et al.
[10]–[12] and Beks [13] have studied the phenomenon of
convection in a vertical high-pressure mercury lamp, supplied
with dc power. Wendelstorf [14] investigated the convection
and the interaction between the plasma of the discharge and
electrodes in a vertical lamp powered in the dc mode.

Manuscript received May 14, 2012; revised November 10, 2012 and May 6,
2013; accepted May 6, 2013. Date of current version July 3, 2013.
The authors are with the Unité d’Étude des Milieux Ionisés et Réactifs,
Institut Préparatoire aux Etudes d’Ingénieurs de Monastir (IPEIM), 5019
Monastir, Tunisia (e-mail: benhamida_mbechir@yahoo.fr).
Color versions of one or more of the figures in this paper are available
online at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TPS.2013.2263199

For these studies, we can take advantage of the axial
symmetry that can be satisfied with a 2-D model. However,
this symmetry is no longer valid in the case of the horizontal
position. That is why the 2-D configuration will no longer be
sufficient, and we must necessarily use a 3-D model, which
naturally requires a longer calculation time.
Indeed, high-pressure mercury lamps are used in horizontal
operating positions. Currently, very few 3-D models have been
developed for these systems [15]–[17]. So, we feel that it
is appropriate to develop a 3-D numerical code based on
solving “fluid” equations governing the plasma in the local
thermodynamic equilibrium (LTE) state. The commercial simulation software COMSOL, which is based on finite technical
elements, is used as the computational platform in which the
applied modules are used for solving the physical models. We
choose to study the influence of the electric current on mass
and energy transport phenomena in both vertical and horizontal
positions.
II. M ODEL
A. Simplifying Assumptions
The discharge lamp is supposed to be in the LTE state.
The fluid is considered as an ideal gas, and all external forces
other than the gravitational force are neglected. The classical
equations of the fluid mechanics described below are written
for the following conditions.
1) A Newtonian fluid, single phase and homogeneous for
3-D flow, is assumed.
2) The flow is assumed to be dominated by diffusion
phenomena and so it is laminar.
3) The terms of the viscous dissipation in the energy
equation are not taken into account.
The plasma column is assumed to be independent of the
electrode properties or the arc attachment to the electrodes.
This assumption is valid as long as the electrode gap is large
enough. In that case, the properties of the plasma column can
be treated without taking the influence of the electrodes into
account [18]. Therefore, in this paper all phenomena at the
electrode surface and electrode regions are omitted. Thus, our
model results can be considered to be valid for a few mean free
paths away from the electrodes. Note that, for short electrode
gaps, this assumption is already problematic.
As was mentioned earlier, in this paper, we use a timedependent 3-D code, thereby solving the coupled momentum,

BEN HAMIDA AND CHARRADA: EFFECT OF ELECTRIC CURRENT BETWEEN VERTICAL AND HORIZONTAL HP MDLs

mass continuity, energy, and electric field equations in the
coordinate system (x, y, z). It can be described by the following
equations.
B. Equations of the Model
Taking into account the above assumptions, the simplified
system of equations is written as follows:
Mass conservation equation
∂
∂
∂
(ρu) +
(ρv) + (ρw) = 0
∂x
∂y
∂z

plasma convective flow. The terms on the right-hand side
represent the plasma heat conduction and the source term,
respectively.
The electrical conductivity, thermal conductivity, and viscosity included in this model are assumed to be described
by the local temperature only, and are calculated by using
the first-order approximation of the gas kinetic theory as
developed by Hirchfelder et al. [19] by assuming a Maxwellian
shape for the electron energy distribution function and the
Lennard–Jones’interatomic potential. The value corresponding
to a mono-atomic ideal gas is used for the specific heat Cp
mentioned by Chase et al. [20].
The transport coefficients calculated in the frame of the firstorder approximation are in agreement with those in the plasma
science literature [2], [21], [22].
An alternative method to calculate the radiation energy
transport term is to replace it by a simple function Urad =
Uemit − Uabs representing the difference between the emitted
and absorbed radiation in the lamp [23], [24].
The calculation of the net emission always involves a
number of approximations which limit the field of application
of each method. More recently, Bouaoun et al. [27] have established a semiempirical formula for calculating the coefficient
of the net issuance in the following form:

c
p
b1
1
a1 exp
− a2 (β − 1)
exp −
Uabs (T ) =
T
Tc
T

c
b2
2
× exp −
exp −
(8)
Tc
T
where Tc and β characterize the parabolic profile of the
temperature. The empirical constants (ai , bi , and ci ) were
determined by using an appropriate program applied to the
dataset temperature (Tc , β) and pressure. The values of the
constants are presented in Table I.
The absorption coefficient for the temperature region below
4000 K was assumed to be proportional to both the mercury
pressure (absorber density) and arc power (radiation flux) as
adopted in [25]. The proportionality constant was selected to
be consistent with the Elenbaas results [1]. The net emission
coefficient Uemit at temperature T has been approximately
expressed by a single exponential [1]
¯

Uemit = Ng C1 e−e V / kT
(7)

where Cp is the specific heat capacity at constant pressure, σ
is the electrical conductivity, E is the electric field, λ is the
gas thermal conductivity, and Urad is the total radiated power
of the arc.
In (7), the first term on the left-hand side is the time rate
of change in energy, while the second term is related to the

(9)

where V¯ is the average excitation potential, Ng is the gas
density, C1 is a constant (C1 = 0.70 × 10−10 ), and k is the
Boltzmann constant.
The state equation of the ideal gas is given by
pM
(10)
ρ=
RT
where M is the atomic mass and R is the ideal gas constant.

Fig. 2. Radial profile of temperature halfway between the electrodes in a
vertical lamp.

TABLE III

III. R ESULTS AND D ISCUSSION

C HARACTERISTICS OF THE L AMP

A. Validation
Interelectrode length (mm)

72

Internal diameter (mm)

18.5

Length of the electrode (mm)

7.25

Diameter of the electrode (mm)

2

C. Boundary Conditions
Because of the complex geometry of the arc tube, a simplified configuration is adopted that is believed to be sufficient
to render the global effects of the convective mode on the arc
plasma behavior. The direction of the gravitational effect for
the lamp in the vertical position is along the z-axis while it
is along the y- axis in the horizontal position. This is shown
in Fig. 1.
The boundary conditions used in this model are summarized
with reference to Fig. 1 and Table II.
The characteristics of the studied lamp are given in Table III.

Since we could not find in the literature any experimental
results concerning high-pressure mercury lamps operating in
horizontal positions, we were obliged to validate our 3-D
model by taking a vertical configuration because the characteristics of the lamp used is very close to those of the lamp
used in [25]. Fig. 2 shows the temperature profile calculated by
our 3-D model, in a midway section between the electrodes,
compared to that measured by Zollweg. The characteristics of
the lamp used by Zollweg are given in Table IV.
We can notice the quite satisfactory agreement between
the calculated and the measured values, thereby justifying the
various assumptions adopted in the 3-D model. We notice,
however, that our results are also in good agreement with the
experimental results of other authors (not reported here) [25].
To test the sensitivity of the 3-D code, we chose to
reproduce the effect of the arc current, first vertically and
then horizontally. These results are shown in the following
paragraphs.

BEN HAMIDA AND CHARRADA: EFFECT OF ELECTRIC CURRENT BETWEEN VERTICAL AND HORIZONTAL HP MDLs

Fig. 4. Temperature profile at the midsection of the lamp between electrodes
for different currents in a vertical lamp.

B. Influence of Arc Current
In this section are presented the results showing the influence of the arc current on the distribution of the temperature
and the transport phenomena. For the pressure and mass of
mercury, respectively, 6 × 105 Pa and 138.5 mg, we vary the
arc current from 1 to 10 A.
1) Spatial Distribution of the Temperature: Fig. 3 shows
the distribution of the temperature for different currents in the
case of a vertical lamp.

From Fig. 3, the arc appears more homogeneous for high
current values. Assuming that the instability of the discharge
is essentially due to convection flow, we conclude that a high
current is a disadvantage for convection.
Fig. 4 shows the radial profiles of temperature midway
between the electrodes for different arc currents for the vertical
lamp. The relative shape of these profiles coincides well with
that found experimentally by Zollweg [25]. This graph also
shows that the hot channel of the arc widens when the current
increases, whereas the central temperature varies only slowly.
This same phenomenon was observed by Stromberg [26].
Fig. 5 shows the distribution of the temperature for different
currents in the case of horizontal lamp.
According to Fig. 5, it is clear that the conductive zone
expanded when the current increased. The “up” zone of the
plasma remains cold and becomes hotter and hotter when the
power injected into the plasma increases.
In reality, more than the half of the mass of mercury initially
introduced into the burner does not participate in the discharge.
For low currents, it is clear that the cold zone of the plasma
has relatively low temperatures, which favors the absorption of
radiation and thereafter the degradation of the efficiency of the
source. With increase in the injected electrical power, this zone
will be increasingly heated and contributes to the discharge,
thereby improving the efficiency of the lamp (Fig. 6).

1700

Fig. 6. Temperature profile at the midsection of the lamp between electrodes
for different currents in a horizontal lamp.

Fig. 7. Radial profiles of the heat conduction flow halfway between electrodes
for different currents in a vertical lamp.

2) Flow of Conduction: The enlargement of the temperature
profile with the increase in the current, on the one hand,
supports the growth of the radial component of the heat flow
to the edges of the tube and, on the other hand, reduces the
same amount at the center of the arc.
The increase in the current at constant pressure implies
an increase in these heat losses, since the losses are directly
related to the value of heat flux on the wall. The increase in
heat flow in the peripheral regions of the discharge with the
current is shown on Fig. 7 for a vertical lamp.
Fig. 8 shows the profiles of the heat conduction flow
halfway between the electrodes for different currents in a
horizontal lamp.
According to Fig. 8, we see that the heat conduction in a
horizontal lamp differs from that in a vertical lamp [12]. We
also note that the heat flow in the “up” zone is becoming
increasingly important when the current increases.
3) Convective Flow: Fig. 9 shows the profiles of the convective flow for various arc currents for a vertical lamp. We note
that this convective flow decreases by increasing the current.
Fig. 10 represents the radial profiles of the convective
flow at halfway between electrodes for different currents in
a horizontal lamp.

IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 41, NO. 7, JULY 2013

Fig. 8. Profiles of the heat conduction flow at the midsection of the lamp
between electrodes for different currents in a horizontal lamp.

Fig. 9. Radial profiles of the heat convective flow halfway between electrodes
for different arc currents in a vertical lamp.

Fig. 10.
Heat convection flow at the midsection of the lamp between
electrodes for different currents in a horizontal lamp.

From Fig. 10, we see that the heat convection flow in the
“up” zone increases more and more while in the “down” zone
it decreases. The maximum shifts from the “down” to the “up”
zone while the current increases.
4) Accumulation of Mercury Behind the Electrodes: Fig. 11
shows the variation of the amount of mercury trapped behind
the electrodes as a function of arc current. We notice that the

BEN HAMIDA AND CHARRADA: EFFECT OF ELECTRIC CURRENT BETWEEN VERTICAL AND HORIZONTAL HP MDLs

1701

electrode, as the convective transfer in the horizontal lamp is
radial. In addition, the current variation is not a determining
factor because the mercury lost behind the electrodes is
insignificant.
IV. C ONCLUSION

Fig. 11.

Mass of mercury trapped behind the electrodes in a vertical lamp.

Fig. 12. Variation of the percentage of mercury accumulated behind the
electrodes as a function of arc current for a pressure of 6 × 105 Pa.

Fig. 13. Mass of mercury behind the electrodes for different currents in a
horizontal lamp.

amount of mercury accumulated behind the electrodes does
not vary much when the current varies.
A possible explanation for this phenomenon is the effect of
heat from electrodes.
From Fig. 12, it is seen that the percentage of the mass
trapped behind the electrodes is an increasing function of the
arc current and this percentage varies only by 7% when the
current goes from 1 to 10 A.
Fig. 13 shows the mass of mercury behind the electrodes
for different currents.
We observe that the mass of mercury lost behind the lower
electrode is almost the same as that hidden behind the higher

This paper presented a 3-D modeling of a high-pressure
discharge plasma. The proposed model was applied to the
study of energy transfer in a mercury discharge lamp powered
by direct and variable current. The designed model made it
possible to solve the coupled system of energy, mass, and
momentum equations, as well as the Laplace equation for the
plasma.
The results obtained from the 3-D model in a vertical configuration of the discharge reproduced perfectly the evolution
characteristics published in the literature. This allowed us to
validate the different assumptions and relations adopted in our
model by comparing our calculations with the corresponding
experimental results. Finally, we examined the effect of electric current on the behavior of the discharge.
The main results obtained for the case of a mercury lamp in
a horizontal position when the current increases are as follows.
1) For low currents, the cold zone of the plasma has
relatively low temperatures, which favors the absorption
of radiation and consequently the degradation of efficiency of the source. With the increase in the injected
electrical power, the zone is more and more heated and
contributes to the discharge, which is likely to improve
the performance of the lamp.
2) Based on the principle that the heat flux is directed from
the hottest zone to the coldest zone, we deduce that
the heat flux of conduction in the “up” area becomes
increasingly important.
3) The heat convection flow in the “up” zone increases
more and more, while that in the “down” zone decreases
and the maximum shifts from the “down” to “up” zone
as well.
4) The mass of mercury lost behind the “up” electrode is
almost the same as that hidden behind the “down” electrode. Therefore, there is neither a cold zone behind the
left electrode nor a hot zone behind the right electrode.
R EFERENCES
[1] W. Elenbaas, The High Pressure Mercury Vapor Discharge. Amsterdam,
The Netherlands: North Holland, 1951.
[2] R. B. Ahmed, S. Kazziz, L. Troudi, K. Charrada, and H. Helali,
“Dynamic thermal and radiative behaviors of a high pressure sodium
lamp plasma,” Phys. Plasmas, vol. 12, no. 8, pp. 083501-1–083501-8,
2005.
[3] K. C. Paul, T. Takemura, T. Hiramoto, M. Yoshioka, and T. Igarashi,
“Three-dimensional modeling of a direct current operated Hg-Ar lamp,”
IEEE Trans. Plasma Sci., vol. 34, no. 2, pp. 254–262, Apr. 2006.
[4] W. Shyy and P. Y. Chang, “Effects of convection and electric field on
thermofluid transport in horizontal high-pressure mercury arcs,” J. Appl.
Phys., vol. 67, no. 4, pp. 1712–1719, 1990.
[5] P. Y. Chang, W. Shyy, and J. T. Dakin, “A study of three-dimensional
natural convection in high-pressure mercury lamps—I. Parametric variations with horizontal mounting,” Int. J. Heat Mass Transf., vol. 33,
no. 3, pp. 483–493, 1990.

Mohamed Bechir Ben Hamida was born in
Moknine, Tunisia, on July 7, 1977. He received
the Diploma of Engineer degree and the D.E.A.
degree in process engineering - chemical engineering
and a Certificate of Higher Education Specialized
in Chemical Engineering and Industrial Chemistry
(C.E.S.S.) from the National School of Engineers,
Gabès, Tunisia, in 2000 and 2002, respectively, and
the Master’s degree and the Ph.D. degree in energy
engineering from the National School of Engineers,
Monastir, Tunisia, in 2005 and 2012, respectively.
Since September 2012, he has been an Assistant Professor with the Graduate
School of Science and Technology (E.S.S.T H.S), Sousse, Tunisia, and a
Researcher with the Unit for the Study of Ionized Gases and Reagents. He
has published five paper and four international communications. His current
research interests include plasma and discharge lamp applications in model
3-D.

Kamel Charrada was born in Sousse, Tunisia, on
February 5, 1967. He received the Master’s degree
in applied physics from High Normal School of
Technical Education (E.N.S.E.T), Tunisia, in 1990,
and the Ph.D. degree in physics science from Paul
Sabatier University, Toulouse, France, in 1995.
He is currently a Professor with the Preparatory Institute for Engineering Studies (I.P.E.I.M),
Monastir, Tunisia, where he is the Director of the
Unit for the Study of Ionized Gases and Reagents.
His current interests include plasma, discharge lamp
applications, and combustion.